Answer:
Option (4)
Step-by-step explanation:
cos(a + b) = cos(a)cos(b) - sin(a)sin(b) [Identity]
Since, sin(a) = [tex]\frac{2}{5}[/tex] for a in quadrant II
Therefore, cos(a) = [tex]\sqrt{1-\text{sin}^2(a)}=\sqrt{1-(\frac{2}{5})^2}[/tex]
cos(a) = [tex]\frac{\sqrt{21}}{5}[/tex]
But cosine of a is negative quadrant II
Therefore, cos(a) = -[tex]\frac{\sqrt{21}}{5}[/tex]
And cos(b) = [tex]\frac{1}{3}[/tex] for b in quadrant IV
Therefore, sin(b) = [tex]\sqrt{1-\text{cos}^{2}b}[/tex]
= [tex]\sqrt{1-(\frac{1}{3})^2}[/tex]
= [tex]\frac{2\sqrt{2}}{3}[/tex]
But sine of an angle is negative in quadrant IV,
Therefore, sin(b) = [tex]-\frac{2\sqrt{2}}{3}[/tex]
Now substituting these values in the identity,
cos(a + b) = [tex](-\frac{\sqrt{21}}{5})(\frac{1}{3})-(\frac{2}{5})(-\frac{2\sqrt{2}}{3})[/tex]
= [tex]-\frac{\sqrt{21}}{15}+\frac{4\sqrt{2}}{15}[/tex]
= [tex]\frac{4\sqrt{2}-\sqrt{21}}{15}[/tex]
Therefore, Option (4) will be the answer.
